String bordism

Revision as of 15:37, 31 March 2011

1 Introduction

${{Stub}} == Introduction == <wikitex>; $String$-bordism or $O\!\left< 8 \right>$-bordism is a special case of a [[B-Bordism|B-bordism]]. It comes from the tower of fibrations below. $$ \xymatrix{BO\!\left< 8 \right>\ar[d] & \ BSpin \ar[d]\ar[r]^{p_1/2}&K({\mathbb Z},4) \ BSO\ar[d]\ar[r]^{w_2}& K({\mathbb Z}/2,2) \ BO\ar[r]^{w_1}& K({\mathbb Z}/2,1) } $$ In each step the lowest homotopy group is killed by the map into the Eilenberg-MacLane spaces. In particular, $BO\left< 8 \right>$ is the homotopy fibre of the map from $BSpin$ given by half of the first Pontryagin class. The name $String$-group is due to Haynes Miller and will be explained below. </wikitex> == The String group == <wikitex>; There are various models for the String group. Since its first three homotopy groups vanish it can not be realized as a finite dimensional Lie group. There are homotopy theoretic constructions which use loop space constructions, that is, inifnite dimensional techniques. A more geometric way to think about $String(n)$ is the following: the String group fibers over the Spin group with fibre $K({\mathbb Z},2)$. One may think of $ K({\mathbb Z},2)$ as the realization of the finite dimensional Lie groupoid $S^1$ viewed as a category with only one object. This way, the String group as an $A_\infty$-space appear as the realization of a smooth extension of $Spin(n)$ by the finite dimensional Lie groupoid $S^1$ (see \cite{Schommer-Pries2010}). A more explicit model for this extension can be found in \cite{Meinrecken03}. </wikitex> == Low dimensional generators == <wikitex>; The following computations of $\Omega_*^{String}$ for $* \leq 16$ comes from \cite{Giambalvo1971|p. 538}. * For $i \leq 6$ the natural map $\Omega_i^{fr} \to \Omega_i^{String}$ is an isomorphism. See [[Framed bordism|framed bordism]] for these groups. * $\Omega_7^{String} = 0$. * $\Omega_8^{String} \cong \Zz \oplus \Zz/2$, generated by the [[Exotic spheres|exotic 8-sphere]] for the 2-torsion and a certain [[Bott manifold]]: see \cite{Laures2004}. * $\Omega_9^{String} \cong \Theta_9/bP_{10} \cong (\Zz/2)^2$, generated by [[Exotic spheres|exotic 9-spheres]]. * $\Omega_{10}^{String} \cong \Zz/6$, generated by an [[Exotic spheres|exotic 10-sphere]]. * $\Omega_{11}^{String} = 0$. * $\Omega_{12}^{String} \cong \Zz$, generated by a 5-connected manifold with signature \times 992$. * $\Omega_{13}^{String} = 0$. * $\Omega_{14}^{String} \cong \Zz/2$, generated by the [[Exotic spheres|exotic 14-sphere]]. * $\Omega_{15}^{String} \cong \Zz/2$, genreated by the [[Exotic spheres|exotic 15-sphere]]. * $\Omega_{16}^{String} \cong \Zz^2$. </wikitex> == Homology calculations == ===Singular homology=== ===$K(1)$-local computations=== ===$K(2)$-local computations=== ===Computations with respect to complex oriented theories=== <wikitex>; The additive structure of the bordism groups is not fully determined yet. Clearly, since $BO\!\left< 8 \right>$ is 7-connected the first 6 bordism groups coincide with the [[Framed bordism|framed bordism groups]]. It is known that only 2 and 3 torsion appears (see \cite{Giambalvo1971}). Indeed, localized at a prime $p>3$, string bordism splits additively into a sum of suspensions of $BP$, although the ring structure is different (see \cite{Hovey2008}). The first 16 bordism groups has been computed by Giambalvo, at the prime 3 the first 32 bordism groups can be found in \cite{Hovey&Ravenel1995}. Further calculations have been done in \cite{Mahowald&Gorbounov1995}. </wikitex> == The structure of the spectrum== <wikitex>; </wikitex> == The Witten genus == <wikitex>; At the end of the 80s Ed Witten were studying the $S^1$-equivariant index of the Dirac operator on a loop space of a k$-dimensional manifold. For compact manifolds a Dirac operator exists if the manifold is Spin. For the loop space $LM$ this would mean that $M$ is $String$. Witten carried the Atiyah-Segal formula for the index over to this infinite dimensional setting and obtained an integral modular form of weight $k$. Nowadays this is called the Witten genus (see \cite{Segal1988}.) The Witten genus can be refined to a map of structured ring spectra $$W: MString \longrightarrow TMF$$ from the Thom spectrum of String bordism to the spectrum $TMF$ of topological modular forms (\cite{Ando&Hopkins&Rezk2006}). This map is also called the $\sigma$-orientation and is 15-connected (see \cite{Hill2008}). The spectrum $TMF$ was developed by Mike Hopkins and Haynes Miller (see \cite{Hopkins2002}.) It is supposed to play the same role for $String$-bordism as $KO$-theory does for $Spin$-bordism. Its coefficients localized away from 2 and 3 are given by the integral modular forms. The map $W$ gives characteristic numbers which together with $KO$ and Stiefel-Whitney numbers are conjectured to determine the $String$ bordism class. Moreover, $tmf$ is supposed to be a direct summand of $MString$ as the orientation map $W$ is shown to be surjective in homotopy (see \cite{Hopkins&Mahowald2002}.) </wikitex> == Characteristic numbers and index theory== <wikitex>; == References == {{#RefList:}} <!-- Please modify these headings or choose other headings according to your needs. --> [[Category:Manifolds]] [[Category:Bordism]]String$-bordism or $O\!\left< 8 \right>$-bordism is a special case of a B-bordism. It comes from the tower of fibrations below.

In each step the lowest homotopy group is killed by the map into the Eilenberg-MacLane spaces. In particular, $BO\left< 8 \right>$ is the homotopy fibre of the map from $BSpin$ given by half of the first Pontryagin class. The name $String$-group is due to Haynes Miller and will be explained below.

2 The String group

There are various models for the String group. However, since its first three homotopy groups vanish it can not be realized as a finite dimensional Lie group. There are homotopy theoretic constructions which involve path spaces, that is, inifnite dimensional techniques. A more geometric way to think about $String(n)$ is the following: the String group fibers over the Spin group with fibre $K({\mathbb Z},2)$. One may think of $K({\mathbb Z},2)$ as the realization of $S^1$ viewed as a smooth category with only one object. This way, the $A_\infty$ space $String(n)$ appears as the realization of a smooth extension of $Spin(n)$ by the finite dimensional Lie groupoid $S^1$ (see [Schommer-Pries2010]). An explicit model for this extension can be found in [Meinrecken03].

3 Low dimensional generators

The following computations of $\Omega_*^{String}$ for $* \leq 16$ comes from [Giambalvo1971, p. 538].

• For $i \leq 6$ the natural map $\Omega_i^{fr} \to \Omega_i^{String}$ is an isomorphism. See framed bordism for these groups.
• $\Omega_7^{String} = 0$.
• $\Omega_8^{String} \cong \Zz \oplus \Zz/2$, generated by the exotic 8-sphere for the 2-torsion and a certain Bott manifold: see [Laures2004].
• $\Omega_9^{String} \cong \Theta_9/bP_{10} \cong (\Zz/2)^2$, generated by exotic 9-spheres.
• $\Omega_{10}^{String} \cong \Zz/6$, generated by an exotic 10-sphere.
• $\Omega_{11}^{String} = 0$.
• $\Omega_{12}^{String} \cong \Zz$, generated by a 5-connected manifold with signature $8 \times 992$.
• $\Omega_{13}^{String} = 0$.
• $\Omega_{14}^{String} \cong \Zz/2$, generated by the exotic 14-sphere.
• $\Omega_{15}^{String} \cong \Zz/2$, genreated by the exotic 15-sphere.
• $\Omega_{16}^{String} \cong \Zz^2$.

4 Homology calculations

4.1 Singular homology

4.2 $K(1)$-local computations

4.3 $K(2)$-local computations

4.4 Computations with respect to complex oriented theories

The additive structure of the bordism groups is not fully determined yet. Clearly, since $BO\!\left< 8 \right>$ is 7-connected the first 6 bordism groups coincide with the framed bordism groups. It is known that only 2 and 3 torsion appears (see [Giambalvo1971]). Indeed, localized at a prime $p>3$, string bordism splits additively into a sum of suspensions of $BP$, although the ring structure is different (see [Hovey2008]). The first 16 bordism groups has been computed by Giambalvo, at the prime 3 the first 32 bordism groups can be found in [Hovey&Ravenel1995]. Further calculations have been done in [Mahowald&Gorbounov1995].

5 The structure of the spectrum

6 The Witten genus

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The Witten genus can be refined to a map of structured ring spectra

$$\displaystyle W: MString \longrightarrow TMF$$

from the Thom spectrum of String bordism to the spectrum $TMF$ of topological modular forms ([Ando&Hopkins&Rezk2006]). This map is also called the $\sigma$-orientation and is 15-connected (see [Hill2008]). The spectrum $TMF$ was developed by Mike Hopkins and Haynes Miller (see [Hopkins2002].) It is supposed to play the same role for $String$-bordism as $KO$-theory does for $Spin$-bordism. Its coefficients localized away from 2 and 3 are given by the integral modular forms. The map $W$ gives characteristic numbers which together with $KO$ and Stiefel-Whitney numbers are conjectured to determine the $String$ bordism class. Moreover, $tmf$ is supposed to be a direct summand of $MString$ as the orientation map $W$ is shown to be surjective in homotopy (see [Hopkins&Mahowald2002].)

7 Characteristic numbers and index theory

References

• [Ando&Hopkins&Rezk2006] Template:Ando&Hopkins&Rezk2006
• [Giambalvo1971] V. Giambalvo, On $\langle 8\rangle$-cobordism, Illinois J. Math. 15 (1971), 533–541. MR0287553 (44 #4757) Zbl 0221.57019
• [Hill2008] M. A. Hill, The String bordism of $BE_8$ and $BE_8\times BE_8$ through dimension 14, (2008). Available at the arXiv:arXiv:0807.2095v1.
• [Hopkins&Mahowald2002] M. Mahowald and M. Hopkins, The structure of 24 dimensional manifolds having normal bundles which lift to $BO[8]$, Recent progress in homotopy theory (Baltimore, MD, 2000), Contemp. Math., 293 (2002), 89–110. MR1887530 (2003b:55007) Zbl 1012.57041
• [Hopkins2002] M. J. Hopkins, Algebraic topology and modular forms, (2002), 291–317. MR1989190 (2004g:11032) Zbl 1031.55007
• [Hovey&Ravenel1995] M. A. Hovey and D. C. Ravenel, The $7$-connected cobordism ring at $p=3$, Trans. Amer. Math. Soc. 347 (1995), no.9, 3473–3502. MR1297530 (95m:55008) Zbl 0852.55008
• [Hovey2008] M. Hovey, The homotopy of $M\text{String}$ and $M\text{U}\langle 6\rangle$ at large primes, Algebr. Geom. Topol. 8 (2008), no.4, 2401–2414. MR2465746 (2009h:55002) Zbl 1165.55001
• [Laures2004] G. Laures, $K(1)$-local topological modular forms, Invent. Math. 157 (2004), no.2, 371–403. MR2076927 (2005h:55003) Zbl 1078.55010
• [Mahowald&Gorbounov1995] M. Mahowald and V. Gorbounov, Some homotopy of the cobordism spectrum $MO\langle 8\rangle$, Homotopy theory and its applications (Cocoyoc, 1993), Amer. Math. Soc. (1995), 105–119. MR1349133 (96i:55010) Zbl 0840.55002
• [Meinrecken03] Template:Meinrecken03
• [Schommer-Pries2010] Template:Schommer-Pries2010
• [Segal1988] G. Segal, Elliptic cohomology (after Landweber-Stong, Ochanine, Witten, and others), Séminaire Bourbaki, Vol. 1987/88, Astérisque No. 161-162 (1988), Exp. No. 695, 4, (1989) 187–201. MR992209 (91b:55005) Zbl 0686.55003